Applications of Linear Equations. Chapter 5

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2 5-2 Applications of Linear Equations Chapter 5

3 5-3 After completing this chapter, you will be able to: > Solve two linear equations in two variables > Solve problems that require setting up two linear equations in two variables > Graph a linear equation in two variables > Perform linear cost-volume-profit and breakeven analysis employing: The contribution margin approach The algebraic approach of solving the cost and revenue functions A break-even chart

4 Solving Two Equations in Two Unknowns 5-4 2x 3y = 6 x + y = x 3y = 6 2 2: 2x + 2y = 4 Subtract y = 10 y = 2 Substitute y=2 into equation 1 2x 3(2) = 6 2x = x = 0

5 Solving Two Equations in Two Unknowns 5-5 2x 3y = 6 x + y = You should always check your answer by substituting the values into each of the equations. x = 0, y = 2 x + y = 2 2 2x 3y = 6 1 LS = = 2 RS = 2 LS = 2(0) -3(2) = -6 Therefore LS = RS RS = -6 Therefore LS = RS x = 0, y = 2

6 Market research for a new product indicates that the product can be sold at $50 per unit. Cost analysis provides the following information: Fixed costs per period = $8640 = FC Variable costs per period = $30. = VC Production capacity per period = 900 units What are Fixed Costs? Fixed costs do not change if sales increase or decrease. > rent > management salaries > some forms of depreciation expense > property taxes 5-6

7 Market research for a new product indicates that the product can be sold at $50 per unit. Cost analysis provides the following information: Fixed costs per period = $8640 = FC Variable costs per period = $30. = VC Production capacity per period = 900 units What are Variable Costs? Variable costs grow in direct proportion to the volume of sales. If unit sales increase by 10%, total variable costs will increase by 10%. > materials costs > direct labour costs in manufacturing > wholesale cost of goods in retailing 5-7

8 Market research for a new product indicates that the product can be sold at $50 per unit. Cost analysis provides the following information: 5-8 Fixed costs per period = $8640 = FC Variable costs per period = $30. = VC Production capacity per period = 900 units Calculate the breakeven point: > in units > as a percent of capacity > in $$

9 5-9 Contribution Margin Approach How much does the sale of an additional unit of a firm s product contribute towards increasing its net income? CM = SP - VC CM = contribution margin SP = unit price NI = net income CR = contribution rate At Breakeven, X = FC / CM CR = CM/SP * 100% At Breakeven, NI = $0 If X = number of units sold

10 5-10 So, in this example, SP = 50 FC = 8640 VC = 30 Capacity = 900 CM = SP - VC = = $20 At Breakeven, X = FC / CM = 8640 / 20 = 432 units As a % of capacity, B/E = # units to B/E / # units at capacity = 432 / 900 *100 = 48% of capacity B/E in $$ = # units to B/E * unit price = 432 * 50 = $21 600

11 The lighting division of Seneca Electric Co. plans to introduce a new street light based on the following accounting information: FC = $3136 Calculate the breakeven point: VC = $157. > in units SP= $185 > as a percent of capacity Capacity = 320 units > in $$ SP -VC CM = 112 / 320 *100 = 35% of capacity B/E = X = FC / CM = 3136 / 28 = 112 units 5-11 B/E = # units to B/E * unit price = 112 * $185 B/E = # units to B/E / # units at capacity = $20 720

12 5-12 Determine the B/E point as a % of capacity if FC are reduced to $2688. FC = $3136 $2688 VC = $157. SP= $185 Capacity = 320 units SP -VC CM B/E = X = FC / CM = 2688 / 28 = 96 units B/E = # units to B/E / # units at capacity = 96 / 320 *100 = 30% of capacity

13 Determine the B/E point as a % of capacity if FC increase to $4588 and VC are reduced to 80% of SP? FC = $3136 $4588 VC = $157..8*185=$148 SP= $185 Capacity = 320 units SP -VC CM B/E = X = FC / CM = 4588 / 37 = 124 units B/E = # units to B/E / # units at capacity = 124 / 320 *100 = 39% of capacity 5-13

14 Determine the B/E point as a % of capacity if SP is reduced to $171? FC = $3136 VC = $157. SP= $185 $171 Capacity = 320 units SP -VC CM B/E = X = FC / CM = 3136 / 23 = units ~ 137 units B/E = # units to B/E / # units at capacity = 137 / 320 *100 = 43% of capacity 5-14

15 Returning to the original situation, find the net income if the lighting division sells 134 street lights. FC = $3136 VC = $157. SP= $185 Capacity = 320 units SP -VC CM Each unit provides a CM of $28 therefore, the company s NI is: # units above B/E * CM = 20 * 28 = $560. NI B/E = X = FC / CM = 3136 / 28 = 112 units 5-15 Is the company operating at a profit or a loss? They are operating at = 20 units above B/E.

16 Remaining with the original situation, what unit sales will generate a net income of $2000? SP -VC CM NI / CM = 2000 / 28 B/E = X = FC / CM = units contributing to NI The business is selling = units ~ 184 units = 3136 / 28 = 112 units When NI is positive, the company is operating above B/E. unit sales must be more than 112 units 5-16

17 5-17 If the company experiences a loss of $336, find the unit sales. SP -VC CM B/E = X = FC / CM = 3136 / 28 = 112 units Loss: the company is operating below B/E. B/E level - (NI) / CM = /28 = The business is operating at 12 units below B/E = 100 units sold

18 5-18 Determine the profit or loss when the company operates at 85% of capacity. FC = $3136 VC = $157. SP= $185 Capacity = 320 units SP -VC CM B/E = X = FC / CM 85% of capacity: = 0.85 * 320 = 272 units = 3136 / 28 = 112 units They are operating at = 60 units above B/E. NI = # units above B/E * CM = 60 * 28 = $1680

19 5-19 Last year Marconi Printing had total sales of $ while operating of 75% of capacity. The total of its variable costs was $ Fixed costs were $ What is Marchoni s breakeven point expressed in dollars of sales? VC = Total VC Total unit sales SP -VC CM = $ = $ Let SP = $1 and X represent the number of $1 units sold. TC = FC + VC = $ X NI = CM*X - FC = 0.60X At B/E, NI = $0 0 = 0.60X - $ B/E X = $ / 0.60 = units

20 5-20 THE END

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